University of Washington
Department of Mathematics Final Examination
MATH 394: Probability I
Summer 2026 — Instructor: Arman Jahangiri

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Let the joint probability density function of random variables \(X\) and \(Y\) be given by \[ f(x,y) = \begin {cases} x^2e^{-x(y+1)}, & x\geq 0,\quad y\geq 0,\\ 0, & \text {elsewhere}. \end {cases} \]

The marginal probability density functions are \[ f_X(x) = \begin {cases} xe^{-x}, & x\geq 0,\\ 0, & \text {otherwise}, \end {cases} \] and \[ f_Y(y) = \begin {cases} \dfrac {2}{(1+y)^3}, & y\geq 0,\\ 0, & \text {otherwise}. \end {cases} \]

(a)

Calculate \[ \operatorname {Cov}(X,Y). \] (4 points)

(b)

Calculate \[ \operatorname {Cov}(3X+1,\,2X-4). \] (3 points)

(c)

Are \(X\) and \(Y\) independent? Justify your answer. (2 points)

(d)

Calculate the moment-generating function of \(X\), namely \[ M_X(t)=E(e^{tX}). \] (5 points)

(e)

Using \(M_X(t)\), calculate \(E(X)\). (2 points)

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Let \(X\) and \(Y\) be two positive independent continuous random variables with probability density functions \(f_1(x)\) and \(f_2(y)\), respectively.

Find the probability density function of \[ U=\frac {X}{Y}. \]

Hint: Let \(V=X\). Find the joint probability density function of \(U\) and \(V\) (Jacobian method), and then calculate the marginal probability density function of \(U\)._______________________________________________________________________

Let \(X\) and \(Y\) be i.i.d. positive random variables. Assume that all expectations appearing below exist and are finite.

For each part below, fill in the appropriate equality or inequality symbol.

Write \(=\) if the two sides are always equal, \(\leq \) if the left-hand side is less than or equal to the right-hand side but not necessarily equal, and similarly for \(\geq \). If no relation holds in general, write \(?\).

Justify each answer.

(a)

\[ E(\log X) \quad \underline {\hspace {1.2cm}} \quad \log (E(X)). \] (3 points)

(b)

\[ P(X\leq Y) \quad \underline {\hspace {1.2cm}} \quad P(X\geq Y). \] (2 points)

(c)

\[ E(XY) \quad \underline {\hspace {1.2cm}} \quad \sqrt {E(X^2)E(Y^2)}. \] (2 points)

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Suppose \[ X_1,\ldots ,X_n \] are independent Bernoulli random variables with unknown success probability \(p\), and define \[ \widehat p = \frac 1n\sum _{i=1}^n X_i. \]

Let \[ \varepsilon >0 \qquad \text {and}\qquad 0<\alpha <1. \]

Using Chebyshev’s inequality, derive a condition on the sample size \(n\) that guarantees \[ P\left (|\widehat p-p|<\varepsilon \right ) \geq 1-\alpha \] for every \[ 0<p<1. \]_____________

Let \(X\) and \(Y\) be independent exponential random variables with common parameter \(\lambda >0\): \[ X,Y\overset {\mathrm {i.i.d.}}{\sim }\operatorname {Exp}(\lambda ), \] with probability density function \[ f(w)= \begin {cases} \lambda e^{-\lambda w}, & w>0,\\ 0, & \text {otherwise}. \end {cases} \]

Define \[ T=X+Y. \]

Using the convolution formula \[ f_T(t) = \int _{-\infty }^{\infty } f_X(x)f_Y(t-x)\,dx, \] find the probability density function of \(T\), including its support.______________

Let \(X,Y,Z\) be jointly continuous with joint probability density function \[ f_{X,Y,Z}(x,y,z) = \begin {cases} x^2e^{-x(1+y+z)}, & x,y,z>0,\\ 0, & \text {otherwise}. \end {cases} \]

The marginal probability density functions are \[ f_X(x) = \begin {cases} e^{-x}, & x>0,\\ 0, & \text {otherwise}, \end {cases} \] and \[ f_Y(y) = \begin {cases} \dfrac {1}{(1+y)^2}, & y>0,\\ 0, & \text {otherwise}, \end {cases} \qquad f_Z(z) = \begin {cases} \dfrac {1}{(1+z)^2}, & z>0,\\ 0, & \text {otherwise}. \end {cases} \]

(a)

Are \(X,Y,Z\) pairwise independent? Justify your answer. (5 points)

(b)

Are \(X,Y,Z\) mutually independent? Justify your answer. (3 points)

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For each part below, select the correct statement of the indicated theorem. Only one choice is correct in each part.

(a)

Weak Law of Large Numbers (WLLN).

Let \(X_1,X_2,\ldots \) be i.i.d. random variables with finite mean \[ E(X_i)=\mu \] and finite variance \[ \operatorname {Var}(X_i)=\sigma ^2>0. \] Let \[ \overline X_n=\frac 1n\sum _{i=1}^n X_i. \]

Which of the following is the conclusion of the Weak Law of Large Numbers? (2 points)

(A)

For every \(\varepsilon >0\), \[ P\left ( |\overline X_n-\mu |>\varepsilon \right )=0 \qquad \text {for every }n. \]

(B)

For every \(\varepsilon >0\), \[ \lim _{n\to \infty } P\left ( |\overline X_n-\mu |>\varepsilon \right ) =0. \]

(C)

\[ P\left ( \lim _{n\to \infty }\overline X_n=\mu \right )=1. \]

(D)

For every \(z\in \mathbb R\), \[ \lim _{n\to \infty } P\left ( \frac {\overline X_n-\mu }{\sigma /\sqrt n} \leq z \right ) = \Phi (z). \]

(b)

Strong Law of Large Numbers (SLLN).

Let \(X_1,X_2,\ldots \) be i.i.d. random variables satisfying the same assumptions as in the Weak Law of Large Numbers, with \[ E(X_i)=\mu \] and finite variance \[ \operatorname {Var}(X_i)=\sigma ^2>0. \]

Which of the following is the conclusion of the Strong Law of Large Numbers? (2 points)

(A)

\[ P\left ( \lim _{n\to \infty }\overline X_n=\mu \right )=1. \]

(B)

For every \(\varepsilon >0\), \[ \lim _{n\to \infty } P\left ( |\overline X_n-\mu |>\varepsilon \right ) =0. \]

(C)

\[ \lim _{n\to \infty } P(\overline X_n=\mu ) =1. \]

(D)

\[ \frac {\overline X_n-\mu }{\sigma /\sqrt n} \xrightarrow []{d} N(0,1). \]

(c)

Central Limit Theorem (CLT).

Let \(X_1,X_2,\ldots \) be i.i.d. random variables with \[ E(X_i)=\mu , \qquad \operatorname {Var}(X_i)=0<\sigma ^2<\infty . \]

Which of the following is the conclusion of the Central Limit Theorem? (2 points)

(A)

\[ \overline X_n \xrightarrow []{P} N(\mu ,\sigma ^2). \]

(B)

\[ P\left ( \lim _{n\to \infty }\overline X_n=\mu \right )=1. \]

(C)

For every \(z\in \mathbb R\), \[ \lim _{n\to \infty } P\left ( \frac {\overline X_n-\mu }{\sigma /\sqrt n} \leq z \right ) = \Phi (z). \]

(D)

For every \(\varepsilon >0\), \[ \lim _{n\to \infty } P\left ( |\overline X_n-\mu |>\varepsilon \right ) =0. \]

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